Discrete Signals
By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH
(Translate to R by Gwangil Seo)
Table of Contents
- 1. Discrete Time Signals
- 2. Complex Signals
- 3. Discrete Sinusoids
- 4. Complex Exponential Signals
1. Discrete Time Signals
A signal \(x[n]\) is a function that maps an independent variable to a dependent variable.
In this course, we will focus on discrete-time signals \(x[n]\) :
- Independent variable is an integer: \(n \in \mathbb{Z}\)
- Dependent variable is a real or complex number: \(x[n] \in \mathbb{R}\ \text{or}\ \mathbb{C}\)
1.1. Plot Real Signals
plot for continuous signals in MATLAB \(\Rightarrow\) plot(type = ‘l’) for continuous signals in R
stem for discrete signals in MATLAB \(\Rightarrow\) plot & arrows & abline for discreate signals in R
\[ x(t) = \text{sin}(2\pi t) \]
t <- seq(0, 2, 0.01)
x <- sin(2 * pi * t)
plot(t, x, type = 'l', lwd = 1, col = 'deepskyblue', ylim = c(-1.1, 1.1), xlab = 't [sec]', ylab = 'x(t)')
\[
x[n] = \text{sin}\left({2\pi\over N} n\right)
\]
N <- 20
n <- 0:(N-1)
x <- sin(2 * pi/N * n)
par(mfrow = c(2, 1), mar = c(4, 4, 1, 1))
plot(n, x, pch = 16, col = 'deepskyblue')
plot(n, x, col = 'deepskyblue', cex = 1.5)
arrows(x0 = n, y0 = 0, x1 = n, y1 = x, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
1.2. Signal Sounds
fs <- 44100 # sampling frequency
N <- 145000 # # of data points
# signal: "Alas, Poor Yorick!"
alas <- R.matlab::readMat("D:/PHM_R/data_files/hamlet.mat")$alas
hamlet <- 2 * alas[1:N]
plot(0:(N - 1) / fs, hamlet, type = 'l', lwd = 1, col = 'deepskyblue', xlab = "", ylab = "")
sound <- function(data, samp.rate) {
W <- tuneR::normalize(tuneR::Wave(data, samp.rate = samp.rate, bit = 16), unit = "16")
tuneR::play(W)
}
sound(hamlet, fs)\[ x[n] = \text{cos}\left( {2\pi \over 200} n \right) \]
# cosine wave
n <- 0:(N - 1)
coswave <- cos(2 * pi / 200 * n)
plot(n / fs, coswave, type = 'l', lwd = 1, col = 'deepskyblue', xlim = c(1, 1.1), xlab = "", ylab = "")
sound(coswave, fs)Chirp
\[ x[n] = 0.3\text{cos}\left({2\pi \over 3000000} n ^ 2\right) \]
# chirp (https://en.wikipedia.org/wiki/Chirp)
chirp <- 0.3 * cos(2 * pi / 3000000 * n ^ 2)
plot(n / fs, chirp, type = 'l', lwd = 1, col = 'deepskyblue', xlim = c(0, 0.4), xlab = "", ylab = "")
sound(chirp, fs)whitenoise
# white gaussian noise
whitenoise <- 0.1 * rnorm(N, 0, 1)
plot(n / fs, whitenoise, type = 'l', lwd = 1, col = 'deepskyblue', xlim = c(0, 0.4), xlab = "", ylab = "")
sound(whitenoise, fs)2. Complex Signals
2.1. ComplexNumbers
\[\text{Complex number = vector in 2D}\]
2.1.1. Adding Complex Numbers
\[ z_1 = a_1 + ib_1, \quad \vec z_1 = \left[ \begin{array} {r} a_1 \\ b_1 \end{array} \right] \\ z_2 = a_2 + ib_2, \quad \vec z_2 = \left[ \begin{array} {r} a_2 \\ b_2 \end{array} \right] \]
\[\begin{align} z &= z_1 + z_2 = (a_1 + a_2) + i(b_1 + b_2) \\ \vec z &= \vec z_1 + \vec z_2 = \left[ \begin{array} {r}a_1\\b_1\end{array} \right] + \left[ \begin{array} {r} a_2 \\ b_2 \end{array} \right] = \left[\begin{array}{r}a_1 + a_2 \\ b_1 + b_2 \end{array} \right] \end{align} \]
2.1.2. Complex Exponential Function (Euler’s Formular)
\[ e^{i\theta} = \text{cos}\theta + i \text{sin}\theta \]
\(\vec z = r\text{cos}\theta + ir\text{sin}\theta \\ \vec z = re^{i\theta} = r(\text{cos}\theta + i\text{sin}\theta) \\ r : \text{magnitude (length)}\\ \theta : \text{phase (angle)}\)
2.1.3. MUltiplying/Dividing Complex Numbers
- Just like an exponential function
\[ z_1 = r_1e^{i\theta_1} \qquad \Longrightarrow \qquad z_1 \dot\\z_2 = r_1r_2e^{i(\theta_1 + \theta_2)} \\ z_2 = r_2e^{i\theta_2} \qquad \Longrightarrow \quad \qquad {z_1 \over z_2} = {r_1 \over r_2}e^{i(\theta_1 - \theta_2)}\ \ \]
2.2. Circular Motion Represented by Complex Number
2.2.1. Geometrical Meaning of \(e^{i\theta}\)
\[ e^{i\theta} = \text{cos}\theta + i\text{sin}\theta \]
\(\quad e^{i\theta}: \text{point on the unit circle with angle of}\ \theta \\ \quad e^{i\theta} = e^{i(\theta + 2\pi)} =\ ...\ =e^{i(\theta + 2\pi k)} =\ ...\\ \qquad \text{static}\)
2.2.2. Geometrical Meaning of \(e^{iwt}\)
\(\quad \theta = wt = 2\pi ft \\ \quad e^{iwt} : \text{rotating on an unit circle with angular velocity of}\ w \\ \qquad \text{dynamic}\)
Question: What is the physical meaning of \(e^{-iwt}\)
2.3. Sinusoidal Functions from Circular Motions (sin & cos)
– Real part (cos term) is the projection onto the \(Re\{\}\) axis
– Imaginary part (sin term) is the projection onto the \(Im\{\}\) axis
| \(\text{projection of}\ e^{jwt}\ \text{onto Re-axis}\) | \(\text{projection of} e^{jwt}\ \text{onto Im-axis}\) |
|---|---|
 |
 |
| \(\text{Re}\left(e^{iwt}\right) = \text{cos}wt\) | \(\text{Im}\left(e^{iwt}\right) = \text{sin}wt\) |
- Animated explanation of \[ \text{cos}wt = {e^{iwt} + e^{-iwt} \over 2} \]
3. Discrete Sinusoids
\[e^{j(wn+\phi)} = \text{cos}(wn + \phi ) + j\text{sin}(wn + \phi)\]
Discrete Sinusoids
\[ x[n] = \text{Acos}(w_0n + \phi)\quad \text{or}\\ x[n] = \text{A}e^{jw_0n + j\phi} \qquad \qquad \ \ \]
Key question: Which one is a higher frequency?
\[w_0 = \pi \quad \text{or} \quad w_0 = {3\pi \over 2}\]
n <- 0:7
x1 <- cos(pi * n)
x2 <- cos(3/2 * pi * n)
par(mfrow = c(2, 1), mar = c(4, 2, 1, 1))
plot(n, x1, col = 'deepskyblue', cex = 1.5, main = expression(x[1]), xlab = "")
arrows(x0 = n, y0 = 0, x1 = n, y1 = x1, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
plot(n, x2, col = 'deepskyblue', cex = 1.5, main = expression(x[2]))
arrows(x0 = n, y0 = 0, x1 = n, y1 = x2, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
3.1. Frequency in Discrete Sinusoids
\[ 0 \rightarrow {1 \over 8 }\pi \rightarrow {2 \over 8 }\pi \rightarrow{8 \over 8 }\pi \]
n <- 0:31
x1 <- cos(0 * pi * n)
x2 <- cos(1/8 * pi * n)
x3 <- cos(2/8 * pi * n)
x4 <- cos(1 * pi * n)
par(mfrow = c(4, 1), mar = c(2.5, 4, 1, 1))
plot(n, x1, col = 'deepskyblue', pch = 16, cex = 1.5, main = expression(x[1]),
xlim = c(0, 31), ylim = c(-1.5, 1.5), ylab = '0')
arrows(x0 = n, y0 = 0, x1 = n, y1 = x1, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
plot(n, x2, col = 'deepskyblue', pch = 16, cex = 1.5, main = expression(x[2]),
xlim = c(0, 31), ylim = c(-1.5, 1.5), ylab = expression(1/8 ~ pi))
arrows(x0 = n, y0 = 0, x1 = n, y1 = x2, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
plot(n, x3, col = 'deepskyblue', pch = 16, cex = 1.5, main = expression(x[3]),
xlim = c(0, 31), ylim = c(-1.5, 1.5), ylab = expression(2/8 ~ pi))
arrows(x0 = n, y0 = 0, x1 = n, y1 = x3, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
plot(n, x4, col = 'deepskyblue', pch = 16, cex = 1.5, main = expression(x[4]),
xlim = c(0, 31), ylim = c(-1.5, 1.5), ylab = expression(8/8 ~ pi))
arrows(x0 = n, y0 = 0, x1 = n, y1 = x4, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
\[ 2\pi \rightarrow {15 \over 8 }\pi \rightarrow {14 \over 8 }\pi \rightarrow{8 \over 8 }\pi \]
n <- 0:31
x5 <- cos(2 * pi * n)
x6 <- cos(15/8 * pi * n)
x7 <- cos(14/8 * pi * n)
x8 <- cos(1 * pi * n)
par(mfrow = c(4, 1), mar = c(2.5, 4, 1, 1))
plot(n, x5, col = 'deepskyblue', pch = 16, cex = 1.5, main = expression(x[1]),
xlim = c(0, 31), ylim = c(-1.5, 1.5), ylab = expression(2 ~ pi))
arrows(x0 = n, y0 = 0, x1 = n, y1 = x5, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
plot(n, x6, col = 'deepskyblue', pch = 16, cex = 1.5, main = expression(x[2]),
xlim = c(0, 31), ylim = c(-1.5, 1.5), ylab = expression(15/8 ~ pi))
arrows(x0 = n, y0 = 0, x1 = n, y1 = x6, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
plot(n, x7, col = 'deepskyblue', pch = 16, cex = 1.5, main = expression(x[3]),
xlim = c(0, 31), ylim = c(-1.5, 1.5), ylab = expression(14/8 ~ pi))
arrows(x0 = n, y0 = 0, x1 = n, y1 = x7, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
plot(n, x8, col = 'deepskyblue', pch = 16, cex = 1.5, main = expression(x[4]),
xlim = c(0, 31), ylim = c(-1.5, 1.5), ylab = expression(8/8 ~ pi))
arrows(x0 = n, y0 = 0, x1 = n, y1 = x8, col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
3.2. Aliasing
t <- seq(0, 10 * 2 * pi, length.out = 300)
x <- sin(t)
y <- cos(t)
par(mfrow = c(2, 1), mar = c(2, 4, 1, 1))
plot(t, x, col = 'deepskyblue', type = 'l', lwd = 1, main = "", ylab = 'sin(t)')
plot(t, y, col = 'deepskyblue', type = 'l', lwd = 1, main = "", ylab = 'cos(t)')
ts <- seq(0, 10 * 2 * pi, length.out = 12)
xs <- sin(ts)
ys <- cos(ts)
par(mfrow = c(2, 1), mar = c(2, 4, 1, 1))
plot(t, x, col = 'deepskyblue', type = 'l', lwd = 1, main = "", ylab = 'sin(t)')
lines(ts, xs, col = 'orange', type = 'o', lwd = 1, lty = 2)
plot(t, y, col = 'deepskyblue', type = 'l', lwd = 1, main = "", ylab = 'cos(t)')
lines(ts, ys, col = 'orange', type = 'o', lwd = 1, lty = 2)
ts <- seq(0, 10 * 2 * pi, length.out = 11)
xs <- sin(ts)
ys <- cos(ts)
par(mfrow = c(2, 1), mar = c(2, 4, 1, 1))
plot(t, x, col = 'deepskyblue', type = 'l', lwd = 1, main = "", ylab = 'sin(t)')
lines(ts, xs, col = 'orange', type = 'o', lwd = 1, lty = 2)
plot(t, y, col = 'deepskyblue', type = 'l', lwd = 1, main = "", ylab = 'cos(t)')
lines(ts, ys, col = 'orange', type = 'o', lwd = 1, lty = 2)
ts <- seq(0, 10 * 2 * pi, length.out = 21)
xs <- sin(ts)
ys <- cos(ts)
par(mfrow = c(2, 1), mar = c(2, 4, 1, 1))
plot(t, x, col = 'deepskyblue', type = 'l', lwd = 1, main = "", ylab = 'sin(t)')
lines(ts, xs, col = 'orange', type = 'o', lwd = 1, lty = 2)
plot(t, y, col = 'deepskyblue', type = 'l', lwd = 1, main = "", ylab = 'cos(t)')
lines(ts, ys, col = 'orange', type = 'o', lwd = 1, lty = 2)
4. Complex Exponential Signals
4.1. Damped Free Oscillation
\[ x[n] = e^{-{n \over N}}e^{j{2\pi \over N}n} \]
# real and image parts
N <- 20
n <- 0:(2 * N - 1)
x <- exp(-n/N) * exp(complex(imaginary = 2 * pi / N * n))1) rectanglar form
par(mfrow = c(2, 1), mar = c(2, 4, 1, 1))
plot(n, Re(x), col = 'deepskyblue', cex = 1, main = '', ylim = c(-1, 1), ylab = 'real')
arrows(x0 = n, y0 = 0, x1 = n, y1 = Re(x), col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
matlines(x = n, y = cbind(1 * exp(-n / N), -1 * exp(-n / N)), type = 'l', lwd = 1, lty = 2, col = 'red')
plot(n, Im(x), col = 'deepskyblue', cex = 1, main = '', ylim = c(-1, 1), ylab = 'real')
arrows(x0 = n, y0 = 0, x1 = n, y1 = Im(x), col = 'deepskyblue', length = 0, angle = 0, lwd = 1)
abline(h = 0)
matlines(x = n, y = cbind(1 * exp(-n / N), -1 * exp(-n / N)), type = 'l', lwd = 1, lty = 2, col = 'red')
library(plotly)
plot_ly(x = ~n, y = ~Re(x), z = ~Im(x))2) polar form \[ x[n] = e^{-{n \over N}}e^{j{2\pi \over N}n} \\ = |x[n]|e^{j\angle x[n]} \]
# polar coordinate
plot_ly(theta = ~Arg(x)/pi*180, r = ~Mod(x), type = 'scatterpolar')